\documentclass{article}
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\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
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\begin{document}
\title{thm\_2EringNorm\_2Espolynom\_\_simplify\_\_def (TMYTDfrZRtQh7sbuSRquHiDjYWMMXWVJJ7g)}
\maketitle
Let $ ty\_2Esemi\_\_ring\_2Esemi\_\_ring : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghg}
\end{equation}
Let $ ty\_2Ecanonical\_2Ecanonical\_\_sum : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}nonempty~(ty\_2Ecanonical\_2Ecanonical\_\_sum~\\
A0)\end{array}
\label{hyp:ogfgpfehjhpfcdfedgbgogpgogjgdgbgmgpfcdfedgbgogpgogjgdgbgmgpfpfdhfhng}
\end{equation}
Let $ ty\_2Ecanonical\_2Espolynom : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ecanonical\_2Espolynom}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfedgbgogpgogjgdgbgmgpfcdfedhahpgmgjhogpgng}
\end{equation}
Let $ c\_2Ecanonical\_2Espolynom\_\_normalize : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ecanonical\_2Espolynom\_\_normalize~\\
A\_27a \in (((ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a)^{{({{ty\_2Ecanonical\_2Espolynom}~{A\_27a}})}})^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedgbgogpgogjgdgbgmgpfcdfedhahpgmgjhogpgngpfpfogpgchngbgmgjgkhfg}
\end{equation}
Let $ c\_2Ecanonical\_2Ecanonical\_\_sum\_\_simplify : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ecanonical\_2Ecanonical\_\_sum\_\_simplify~\\
A\_27a \in (((ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a)^{{({{ty\_2Ecanonical\_2Ecanonical\_\_sum}~{A\_27a}})}})^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedgbgogpgogjgdgbgmgpfcdfedgbgogpgogjgdgbgmgpfpfdhfhngpfpfdhjgngahmgjgggjh}
\end{equation}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgbgogpgogjgdgbgmgpfcdfedhahpgmgjhogpgngpfpfdhjgngahmgjgggjh}
We define $\mname{c\_2Ecanonical\_2Espolynom\_\_simplify}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0sr \in {({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})} .  {{\uniddcgcg} V1x \in {({{ty\_2Ecanonical\_2Espolynom}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{c\_2Ecanonical\_2Ecanonical\_\_sum\_\_simplify}~{A\_27a}})}}~{V0sr}})}}~{({{{ap}~{({{{ap}~{({{c\_2Ecanonical\_2Espolynom\_\_normalize}~{A\_27a}})}}~{V0sr}})}}~{V1x}})}})}}}}$.
\end{definition}
Let $ ty\_2Ering\_2Ering : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ering\_2Ering}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfechjgoghgpfcdfechjgoghg}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_RM : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_RM~A\_27a \in (((\\
A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfne}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_RP : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_RP~A\_27a \in (((\\
A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfaf}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_R1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_R1~A\_27a \in (A\_27a^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfbd}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_R0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_R0~A\_27a \in (A\_27a^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfad}
\end{equation}
Let $ c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a \in (((((ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a)^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}})^{{A\_27a}})^{{A\_27a}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfgngjgpfpfchjgoghgpfcdfechfgdgpgchegehjhahfgpfcdfedhfgngjgpfpfchjgoghg}
\end{equation}
\begin{definition}\label{def:dgpfcdfechjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfpggg}
We define $\mname{c\_2Ering\_2Esemi\_\_ring\_\_of}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{ty\_2Ering\_2Ering}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring}~{A\_27a}})}}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_R0}~{A\_27a}})}}~{V0r}})}})}}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_R1}~{A\_27a}})}}~{V0r}})}})}}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RP}~{A\_27a}})}}~{V0r}})}})}}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RM}~{A\_27a}})}}~{V0r}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechjgoghgoepgchngpfcdfechpfpfdhahpgmgjhogpgngpfpfdhjgngahmgjgggjh}
We define $\mname{c\_2EringNorm\_2Er\_\_spolynom\_\_simplify}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{ty\_2Ering\_2Ering}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ecanonical\_2Espolynom\_\_simplify}}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Ering\_2Esemi\_\_ring\_\_of}}~{A\_27a}})}}~{V0r}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechjgoghgoepgchngpfcdfechpfpfdhahpgmgjhogpgngpfpfogpgchngbgmgjgkhfg}
We define $\mname{c\_2EringNorm\_2Er\_\_spolynom\_\_normalize}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{ty\_2Ering\_2Ering}~{A\_27a}})} .  {({{{ap}~{({{c\_2Ecanonical\_2Espolynom\_\_normalize}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Ering\_2Esemi\_\_ring\_\_of}}~{A\_27a}})}}~{V0r}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechjgoghgoepgchngpfcdfechpfpfdgbgogpgogjgdgbgmgpfpfdhfhngpfpfdhjgngahmgjgggjh}
We define $\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_simplify}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{ty\_2Ering\_2Ering}~{A\_27a}})} .  {({{{ap}~{({{c\_2Ecanonical\_2Ecanonical\_\_sum\_\_simplify}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Ering\_2Esemi\_\_ring\_\_of}}~{A\_27a}})}}~{V0r}})}})}}}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfechjgoghgoepgchngpfcdfedhahpgmgjhogpgngpfpfdhjgngahmgjgggjhpfpfegfggg}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (ty\_2Ering\_2Ering~\\
A\_27a) .  ({\unicdcdadad} V1x \in (ty\_2Ecanonical\_2Espolynom~A\_27a) .  ((ap~\\
(ap~(\mname{c\_2EringNorm\_2Er\_\_spolynom\_\_simplify}~A\_27a)~V0r)~V1x)=\\
(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_simplify}~A\_27a)~\\
V0r)~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_spolynom\_\_normalize}~A\_27a)~V0r)~\\
V1x)))))\end{array}$$
\end{theorem}
\end{document}
